In 1863 Jo Shelby raided through Missouri. They didn't exactly travel in a straght line.
~1500 miles / 41 days = 36+ miles per day. Foote vol. 2, pp. 776-778.
That's over 500 miles a turn. About one region a day. More if they had been going in a straight line. It is an outlier, but it does show a speed that cavalry was capable of maintaining if they were pushing very hard.
For the Union, Grierson's raid across Mississippi was ~400 miles in 15 days, with 5 days off to rest in the middle of it. It was more of a straight line than Shelby's raid, from LaGrange, TN to Baton Rouge. Also Wilson's 1865 "raid" from Eastport, MS to Macon, GA ... 480 miles, 25 days, with 8 days off at Selma. This was 13,500 cavalry travelling in three columns, mostly across muddy hills, and fighting two battles along the way.
In civilian terms, for most terrain:
10 miles / day = real easy
14 miles / day = average travelling speed
20 miles / day = vigorous
30 miles / day = pushing it
35+ miles / day = horse-killing speed
Many CW operations were carried out at horse-killing speed, making substitutions from captured or impressed horses.
This topic has been discussed several times before, most recently
here.